Properly discontinuous group action

Orbit space of a properly discontinuous group action

Let be a (discrete) group acting continuously and properly discontinuously on a topological space , and let be the orbit space with the quotient topology and projection . Then is a covering.¹ topology

Proof

Let . Since acts properly discontinuously, has an open neighbourhood such for any with . Then is an evenly covered open neighbourhood of : and is a homeomorphism for each , since it is surjective and continuous (by construction), injective (no to points in lie in the same orbit by proper continuity), and open (in fact is open, since open implies open for all , so is open and thus is open). Thus every has an evenly covered neighbourhood, so is a covering of the orbit space with itself.

Properties

• A stronger result is obtained for the Orbit space of a properly discontinuous effective group action: is itself the deck transformation group.
• If is simply connected and is a Universal covering.

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Footnotes

1. 2010, Algebraische Topologie, pp. 81–82 ↩